This invention relates to the determination of the concentration of a multi-component solid/liquid or liquid/liquid mixture. It can be applied to any situation where the density of such a mixture is known or can be determined. This invention provides a means for improving the control of a continuous process which handles such a mixture and is dependent on the mixture""s concentration, and can thus assist manufacturers in effectively monitoring and operating their processes. Data generated by this invention can be used to: control a known process fluid to a targeted concentration, for instance in a paper coating process; verify on-line the accuracy of batchwise or continuous ratio blends in the manufacture of such products as food products (ketchup, mayonnaise, syrup), personal care products (skin cream, shampoo), pharmaceutical products, paints, petroleum blends, and the like; and eliminate the excessive empirical work necessary with density monitoring process control systems.
The solute content of a solid/liquid or liquid/liquid mixturexe2x80x94commonly expressed as the mixture""s concentrationxe2x80x94can be determined from the mixture""s true density utilizing a relationship that exists between the mixture""s concentration and density. This relationship consists of a linear correlation between a mixture""s concentration and density, which is unique to each mixture. Stated generally, a unique, linear relationship exists for any solute/solvent mixture (solution or slurry).
Defined hereinbelow is an improvement to the two inaccurate, traditional approaches relating concentration to density. The first, the Non-linear Model, assumes that the solute is completely insoluble in the solvent. The second, the Linear Model, is based on a soluble solute. The present improved relationship is referred to as the CONCENTRATION-DENSITY MODEL. This model allows for a theoretical determination of the concentration-density relationship for a multi-component solid/liquid or liquid/liquid mixture. Included in the Concentration-Density Model is a new concept referred to as ADDITIVE VOLUME COEFFICIENT (AVC). This concept compensates the Non-linear Model for the fact that the net volume of a mixture does not always equal the sum of the volumes of each component.
As described in detail herein, this improved Concentration-Density Model provides fluid-handling manufacturers with a method for accurately determining a mixture""s concentration on-line with the aid of current density measurement instrumentation. The Concentration-Density Model of the present invention allows for accurate concentration determination in manufacturing scenarios where such measures were previously impractical. These concentration measurements can then be used to control the manufacturing process.
It is common for a solid/liquid mixture""s concentration to be determined in a laboratory by measuring the weight of a sample both before and after evaporating the liquid phase of the mixture. This approach can be very accurate, but must occur off-line which results in a significant time delay between the time of sampling and the time of measurement. This time delay decreases the number of manufacturing applications where this measurement is useful.
Other methods, either off-line or on-line, determine concentration indirectly. A property of a solid/liquid or liquid/liquid mixture (e.g., density, gamma radiation absorption, and so on) is empirically correlated to a mixture""s concentration. The mixture""s concentration can then be calculated from a measurement of that property (on-line or off-line).
All of these methods, however, face certain challenges. For the sake of accuracy, each mixture to be measured requires the empirical determination of the relationship between the mixture""s concentration and density. When many mixtures are involved, this can result in a great deal of upfront effort. To minimize this upfront effort, one of two models relating a mixture""s concentration to its density is commonly used.
NON-LINEAR MODEL (Insoluble, Two-component System):
Modeling the case of a mixture (M) where the solute (S) is completely insoluble in the solvent (L), the volumes (V) are additive:
VM=VS+VLxe2x80x83xe2x80x83Eq.1
Therefore:                                           m            M                                ρ            M                          =                                                            m                M                            ·                              x                S                                                    ρ              S                                +                                                    m                M                            ·                              (                                  1                  -                                      x                    S                                                  )                                                    ρ              L                                                          Eq.  2-1            
where: m=mass, and, xcfx81= density;
or,                               1                      ρ            M                          =                                            x              S                                      ρ              S                                +                                    1              -                              x                S                                                    ρ              L                                                          Eq.  2-2            
Solving Eq. 2-2 for the mixture""s solute content, xs:                               x          S                =                                            ρ              S                        ⁡                          (                                                ρ                  M                                -                                  ρ                  L                                            )                                                          ρ              M                        ⁡                          (                                                ρ                  S                                -                                  ρ                  L                                            )                                                          Eq.  3            
Eq.3 can be rewritten as:                               x          S                =                                            ρ              S                                      (                                                ρ                  S                                -                                  ρ                  L                                            )                                -                                                                      ρ                  S                                ·                                  ρ                  L                                                            (                                                      ρ                    S                                    -                                      ρ                    L                                                  )                                      ⁢                          xe2x80x83                        ⁢                          (                              1                                  ρ                  M                                            )                                                          Eq.  4            
Therefore, on a plot comparing the mixture""s concentration to the inverse of it""s density:       Slope    =                            ρ          S                ·                  ρ          L                            (                              ρ            S                    -                      ρ            L                          )              ;      xe2x80x83    ⁢            y      ⁢              -            ⁢      Intercept        =                  ρ        S                    (                              ρ            S                    -                      ρ            L                          )            
LINEAR MODEL (Soluble, Two-component System):
Modeling the case of a mixture (M) where the solute is considered soluble in the solvent, it is assumed that the density of the mixture varies linearly from the density of the pure solvent (L) to the density of the pure solute (S), based on the mass ratio of the two components.
This model is expressed as:                               x          S                =                                            ρ              M                        -                          ρ              L                                                          ρ              S                        -                          ρ              L                                                          Eq.  5-1            
which can be re-written as:                                           x            S                    =                                    -                                                ρ                  S                                                                      ρ                    S                                    -                                      ρ                    L                                                                        +                                          ρ                M                                                              ρ                  S                                -                                  ρ                  L                                                                    ⁢                  xe2x80x83                                    Eq.  5-2            
Therefore, on a plot comparing the mixture""s solids content to the it""s density:       Slope    =          1                        ρ          S                -                  ρ          L                      ;      xe2x80x83    ⁢            y      ⁢              -            ⁢      Intercept        =          -                        ρ          L                                      ρ            S                    -                      ρ            L                              
In both of the above, the formula components are defined as follows:
VM=Total System Volume (Volume of Mixture)
VS=Volume of the Solute
VL=Volume of the Solvent
mM=Total System Mass (Mass of Mixture)
xS=Mass Fraction of the Solute
xL=Mass Fraction of the Solvent
xcfx81M=Density of Mixture
xcfx81S=Absolute Density (not Bulk Density) of Solute
xcfx81L=Density of Solvent
However, both of these common models used make erroneous assumptions. In the Non-Linear Model, it is assumed that the volumes of the components are completely additive, meaning that the components are completely insoluble in each other. This is very rarely the case. Most real world cases employ a solution of soluble or partially soluble solutes. In these cases, this model tends to overestimate the solution""s concentration. In the Linear Model, the assumption fails because it does not compensate for the molecular interactions between the solute and the solvent. In either case, the assumptions often introduce enough error to render the results useless, as shown in Table 1.
The present invention provides a new method for predicting the concentration of a solid/liquid or liquid/liquid mixture by use of the mixture""s true density. This method makes use of a model, referred to herein as the Concentration-Density Model, which is an improvement over current methods of relating a mixture""s concentration and density. This model introduces a novel concept referred to as the Additive Volume Coefficient (AVC), which reflects the change in volume that occurs after dissolving or mixing a solute into a solvent. The AVC is an important concept and provides this method with advantages over current technology.
The present invention provides a more accurate measurement of concentration than current technologies, and applies to a wider range of applications. As a result, the present invention, when coupled with an on-line measurement of solution density, provides accurate, continuous, real time feedback of a process fluid""s concentration. This measurement is valuable to various industries in that it can assist manufacturers in effectively monitoring and operating their processes. Specifically, this data can be used, based on the manufacturing process, to do such things as:
1) control a known process fluid to a targeted concentration, e.g. in a paper coating process,
2) verify on-line the accuracy of batchwise or continuous ratio blends such as food products (ketchup, mayonnaise, syrup), personal care products (skin cream, shampoo), paints, petroleum blends, etc., and
3) eliminate excessive empirical work necessary with density monitoring process control systems.
The present invention is an improvement in that it more accurately converts a measured mixture density (xcfx81M) to concentration (m) through the formula:   m  =            1              (                  1          -                                    ρ              L                        ⁢                                          ∑                                  i                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                                                                    k                    i                                    ⁢                                      x                    i                                                                                        (                                          ρ                      S                                        )                                    i                                                                    )              -                            ρ          L                          (                      1            -                                          ρ                L                            ⁢                                                ∑                                      i                    =                    1                                    n                                ⁢                                  xe2x80x83                                ⁢                                                                            k                      i                                        ⁢                                          x                      i                                                                                                  (                                              ρ                        S                                            )                                        i                                                                                )                    ⁢              (                  1                      ρ            M                          )            
wherein xcfx81L is the (temperature-dependent) density of the solvent, ki is the Additive Volume Coefficient for each solute, xi is the weight-% dry for each solute, (xcfx81S)i is the (temperature-dependent) density of each solute, and xcfx81M is the (temperature-dependent) density of the mixture.